Comment.Math.Univ.Carolinae 43,1 (2002) On algebra homomorphisms in complex almost f-algebras Abdelmajid Triki Abstract. Extensions of order b

Transcription

1 Comment.Math.Univ.Carolinae 43,1 (2002) On algebra homomorphisms in complex almost f-algebras Abdelmajid Triki Abstract. Extensions of order bounded linear operators on an Archimedean vector lattice to its relatively uniform completion are considered and are applied to show that the multiplication in an Archimedean lattice ordered algebra can be extended, in a unique way, to its relatively uniform completion. This is applied to show, among other things, that any order bounded algebra homomorphism on a complex Archimedean almost f- algebra is a lattice homomorphism. Keywords: vector lattice, order bounded operator, lattice ordered algebra, f-algebra, almost f-algebra Classification: 06F20, 06F25 1. Introduction In this paper, we show that any order bounded linear operator from an Archimedean vector lattice A into a uniformly complete vector lattice B has a unique order bounded extension to the relatively uniform completion of A. As an application, we show that the multiplication in an Archimedean lattice ordered algebra A can be extended, in a unique way, into a lattice ordered algebra multiplication on μ A, the relatively uniform completion of A, insuch a manner that A becomes a sub algebra of μ A. Moreover, μ A is an f-algebra (respectively almost f-algebra, d-algebra) whenever A is an f-algebra (respectively almost f-algebra, d-algebra). In Section 4 we are mainly concerned with generalizing a theorem, due to E. Scheffold (cf. [8, Theorem 2.2]), which states that if A is a Banach almost f-algebra (FF-algebra in his terminology), then any order bounded multiplicative functional on A C, the complexification of A, is a lattice homomorphism. We show in a different manner that in fact the latter result holds for an arbitrary order bounded algebra homomorphism between A C and B C respectively the complexifications of the Archimedean almost f-algebras A and B, provided B is semiprime. Also, a Nagasawa-like theorem is proved for complex f-algebras in order to characterize algebra homomorphisms. 2. Preliminaries For unexplained terminology and the basic results on vector lattices and f- algebras we refer to [4], [7] and [9]. All vector lattices and lattice ordered algebras

2 24 A. Triki under consideration are supposed to be Archimedean and the only topology we consider on these spaces is the relatively uniform topology (cf. [4, Sections 16 and 63]). If ^A is the Dedekind completion of the Archimedean vector lattice A, then A, μ the closure of A in ^A with respect to the relatively uniform topology, is a relatively uniformly complete vector lattice which is, with respect to ([6, Definition 2.12]), the relatively uniform completion of A. The real algebra A is called a lattice ordered algebra if A is a vector lattice with the following property (i) ab 2 A + for all a; b 2 A +. The lattice ordered algebra A is called an almost f -algebra if A verifies the following property: (ii) ab = 0 for all a; b 2 A such that a ^ b =0. The lattice ordered algebra A is called an f-algebra if A verifies the following property: (iii) a ^ b = 0 implies ac ^ b = ca ^ b = 0 for all c 2 A +. The lattice ordered algebra A is called a d-algebra if A verifies the following property: (iv) a ^ b = 0 implies that c(a ^ b) =ca ^ cb and (a ^ b)c = ac ^ bc for all c 2 A +. The lattice ordered algebra A is said to be semiprime if 0 is the only nilpotent element ofa. The linear mapping T defined on the vector lattice A with values in the vector lattice B is called order bounded (notation T 2 L b (A; B) ort 2 L b (A) ifa = B) if T maps order intervals into order intervals. The mapping T 2 L b (A) is said to be an orthomorphism if it follows from jaj^jbj = 0 that jt (a)j^jbj = 0. The collection Orth(A) of all orthomorphisms on A is, with respect to the usual vector spaces operations and composition as multiplication, an Archimedean f-algebra with the identity mapping I A on A as a unit element. If A is a semiprime Archimedean f-algebra, then the mapping ρ : A! Orth(A), which assigns to a 2 A the operator T a defined on A by T a (b) =ab for all b 2 A, is an injective f-algebra homomorphism. Throughout this paper, a semiprime Archimedean f-algebra will be identified with ρ(a) in Orth(A). If A is in addition relatively uniformly complete then A verifies the so-called Stone condition"; i.e., a ^ I 2 A holds for all a 2 A +. We shall denote by A b the subalgebra of all bounded elements in A, i.e., A b = fa 2 A : jaj» I A ; 2 R + g: Let A be an Archimedean semiprime f-algebra, as agreed upon, we consider A as a sub algebra of Orth(A) and we denote by A C = A + ia the complexification of A. If, in addition, A is relatively uniformly complete then A C and Orth(A) C are complex f-algebras (cf. [1, Section 5]).

3 On algebra homomorphisms in complex almost f-algebras 25 Note, in this connection, that even if A is not relatively uniformly complete, A C can be regarded as a sub algebra of the complex f-algebra Orth( A)C μ and hence jaj exists in Orth(A) C for all a 2 A C. The linear mapping T = T 1 + it 2 : A C! B C is called order bounded if T 1 ;T 2 2 L b (A; B). T is said to be contractive if jt (a)j 2»jT(a)j whenever jaj 2» jaj or equivalently if jt (a)j»i B whenever jaj»i A (the absolute value is taken, if necessarily in Orth(A) C or in Orth( B)C μ ). A bijective operator T is said to be bicontractive if T and T 1 are contractive. 3. An extension theorem Let A and B be Archimedean vector lattices and assume, in addition, that B is relatively uniformly complete and let T 2 L b (A; B). In this section we show that T has a unique extension into an element T 0 2 L b ( μ A; B), where μ A is the relatively uniform completion of A. Moreover, we use this result to prove that the relatively uniform completion of an Archimedean lattice ordered algebra is again a lattice ordered algebra. To a given 0» T 2 L b (A; B) we associate the mappings U and V defined on μ A (the relatively uniform completion of A) with values in b B, the Dedekind completion of B, as follows: U(x) = supft (a) :a 2 A; a» x for all x 2 μ Ag; V (x) = infft (a) :a 2 A; x» a for all x 2 μ Ag. In the following lemma we collect some properties of U and V, the easy proof of which we leave to the reader. Lemma 3.1. (1) U and V are increasing mappings and U(x)» V (x) for all x 2 A. μ (2) U(a) =V (a) =T (a) for all a 2 A. (3) U(x + y) U(x)+U(y) and V (x + y)» V (x)+v(y) for all x; y 2 A. μ (4) U( x) = U(x) for all 0» 2 R and U( x) = V (x) for all <0. Next we show that U = V and that U is a positive extension of T to μ A. Lemma 3.2. U 2 L b ( μ A; b B) and it is the (unique) positive extension of T to μ A. Proof: Since A is relatively uniformly dense in μ A and since U(a) =V (a) for all a 2 A, in order to show that U(x) =V (x) for all x 2 μ A, it is therefore sufficient to prove that U and V are relatively uniformly continuous on μ A. To this end, let fx n : n =1; 2;:::g be a relatively uniformly convergent sequence in μ A with limit x. Then, there is v 2 A + such that, for every ">0 there exists a natural number N " such that jx n xj»"v for all n N ", i.e., x "v» x n» x + "v. We have to show that U(x n ) and V (x n ) converge respectively to U(x) and V (x). U(x "v) = supft (a) :a 2 A; a» x "vg. If we put b = a + "v then we have

4 26 A. Triki b 2 A and b» x. It follows that Hence, ft (a) :a 2 A; a» x "vg = ft (b "v) :b 2 A; b» xg: U(x "v) = supft (b) "T (v) :b 2 A; b» xg = supft (b) :b 2 A; b» xg "T (v) = U(x) et (v): Analogously we have that U(x + "v) =U(x)+"T (v). It follows from Lemma 3.1(1) that U(x) "T (u)» U(x n )» U(x)+"T (v): In other words, U(x n ) converges relatively uniformly to U(x) and hence U is relatively uniformly continuous. Similarly, V is relatively uniformly continuous on A μ and it follows that U(x) = V (x) for all x 2 A. μ From Lemma 3.1(2), (3) and (4) we deduce easily that U is a positive linear mapping in L b ( A; μ B) b which extends T to A. μ Finally it is not difficult to see that U is the unique positive extension of T to A. μ This completes the proof. Λ Now, let T be an element in L b (A; B). T can be regarded as an element of L b (A; B). b Hence, there exist two positive elements T 1 and T 2 in L b (A; B) b such that T = T 1 T 2. By the preceding lemma, T 1 and T 2 have unique positive extensions T1 0 and T 2 0 to A μ the relatively uniform completion of A. Hence, T 0 = T1 0 T L b( A; μ B) b is the unique order bounded extension of T to A. μ Moreover, the relatively uniform continuity oft implies that the range of T 0 is contained in B since B is relatively uniformly closed in B. b Summarizing, we have the following theorem: Theorem 3.3. Let A and B be two Archimedean vector lattices. Assume in addition that B is relatively uniformly complete. Then any order bounded linear operator T : A! B has a unique order bounded linear extension T 0 : A μ! B, where A μ is the relatively uniform completion of A. Let A be an Archimedean lattice ordered algebra and let a 2 A +. If we define T a : A! A μ by Ta (y) = ay, then T a 2 L b (A; A) μ and hence, by the preceding theorem, T a extends to T 0 a 2 L b( A). μ For a fixed y 2 ( A) μ + we put ay = T a (y). Now, for a fixed y 2 ( A) μ + we define R y : A! A μ by Ry (a) = Ta(y). 0 Then R y 2 L b (A; A) μ and hence, Ry extends into Ry 0 2 L b( A). μ If for arbitrary x; y 2 ( A) μ + we define on A μ amultiplication by putting xy = Ry(x), 0 then it is an easy task to verify that this multiplication is the unique

5 On algebra homomorphisms in complex almost f-algebras 27 lattice ordered algebra multiplication in A μ which extends the multiplication in A in such a manner that A becomes a subalgebra of A μ with respect to this multiplication. Moreover, it is easy to verify that such amultiplication is an f- algebra (respectively, almost f-algebra, d-algebra) multiplication whenever A is an f-algebra (respectively, almost f-algebra, d-algebra). Thus we obtain the following theorem: Theorem 3.4. Let A be an Archimedean lattice ordered algebra. Then the multiplication in A can be extended in a unique way into a lattice ordered algebra multiplication on A μ in such a manner that A becomes a sub algebra of A. μ Moreover, A μ is an f-algebra (respectively almost f-algebra, d-algebra) whenever A is an f-algebra (respectively almost f-algebra, d-algebra). 4. The main results Let A and B be Archimedean semiprime f-algebras with unit elements e A and e B. The linear operator T : A C! B C is contractive ifjt (a)j»e B whenever jaj»e A. In order to make things surveyable, we first show that if T 2 L b (A C ;B C ) is contractive and verifies T (e A )=e B, then T is positive (i.e., T (a) 2 B + for all a 2 A + ). To this end, we need the following inequalities: (Λ) IfT 2 L b (A C ;B C ) and u 2 A + then jt (u)j»jt j(u)» supfjt (a)j : a 2 A C ; jaj»ug (cf. [9, Lemma 92.5 and Theorem 92.6]). Proposition 4.1. If T 2 L b (A C ;B C ) is contractive and verifies T (e A ) = e B, then T is positive. Proof: Note first that there exist T 1 ;T 2 2 L b (A; B) such that T = T 1 + it 2. So, T 1 (e A )=e B since T (e A )=e B. Consider, now, T as an element ofl b (A C ; b B C ). Then, the contractivity of T together with the inequalities (Λ) imply that jt j(e A )=e B = T 1 (e A ): It follows from the fact that T 1»jT j, that 0» (jt j T 1 )(a)» (jt j T 1 )(e A )=0 for all a 2 A + verifying a» e A : Consequently, the restriction of jt j T 1 to A b is the null operator on A b. The relatively uniform continuity of jt j T 1 together with the relatively uniform density ofa b in A imply now that jt j T 1 is the null operator on A, i.e., jt j = T 1. Finally, the identity jt (a)j 2 = T 1 (a) 2 + T 2 (a) 2 for all a 2 A + ([1, Theorem 5.2]) already implies that T 2 = 0 and hence jt j = T, which is the desired result. Λ Before proving our main theorem we need the following lemma

6 28 A. Triki Lemma 4.2. If (a + ffi A ) 2 A # C verifies ja + ffi Aj»I A then jffj»1. Proof: It follows from the inequalities jaj jffji A»ja + ffi A j»i A and jffji A jaj» ja + ffi A j» I A that (jffj 1)I A» jaj» (1 + jffj)i A. As a consequence of the Stone condition we have that jffj» 1. Indeed, jffj > 1 would lead to I A < (jffj 1) 1 jaj and hence I A =(jffj 1) 1 jaj^i A 2 A, contradictory to our assumption that I A =2 A. Hence, we necessarily have jffj»1. Λ We are now in a position to prove the main result of this paper which extends Theorem 5.3 of [3], proved in the real case under the additional assumption that A has a unit element. Theorem 4.3. Let A and B be Archimedean semiprime f-algebras. If T : A C! B C is an order bounded algebra homomorphism then T is a lattice homomorphism (i.e., T is real and the restriction of T to A is a lattice homomorphism). Proof: First, we show that T is contractive. To this end, let a 2 A C be such that jaj»i A. This implies that 0»ja n j»jaj for n =1; 2;:::,sowehave that ja n j 2 [0; jaj] and it follows from the order boundedness of T that there exists b 2 B + such that jt (a n )j = jtaj n» b for all n. Now, if we put then h is positive and we have jt (a)j_i B = I B + h 0»jT (a n )j_i B =(jt (a)j_i B ) n =(I B + h) n» b _ I B for all n; from this we deduce that 0» I B + nh» b _ I B for all n: Multiplying these inequalities by 1 n we find 0» 1 n I B + h» 1 n (b _ I B) for all n and so 0» h» 1 n (b _ I B I B ): Therefore, it follows from the Archimedean assumption that h = 0. Consequently jt (a)j»i B and hence T is contractive. Now we consider two cases. First case: A has a unit element e A.

7 On algebra homomorphisms in complex almost f-algebras 29 In this case T (e A ) is an idempotent inb. Moreover, it follows from the fact that T (a) =T (ae A )=T (a)t (e A ) for all a 2 A that T (a) is an element of the band F generated by T (e A )inb. F is an f-algebra with unit element T (e A ), so if we consider T as an algebra homomorphism from A C into B C it follows from Proposition 4.1 that T is positive and hence T is a lattice homomorphism (see e.g. [3, Section 5]). Second case: A does not possess a unit element. Observe first that if A μ is the relatively uniform completion of A then it follows from Theorem 3.3 that T has a unique order bounded linear extension T 0 to AC μ with values in BC μ. Moreover, it takes a little effort to verify that T 0 is an algebra homomorphism. So, we shall assume without loss of generality that A and B are relatively uniformly complete. Assume now that A does not possess a unit element and let A b be the f-algebra of all bounded elements in A. Since T is contractive, the restriction of T to (A b ) C (which we shall denote again by T )isa contractive algebra homomorphism from (A b ) C into B C. Consider now T # the extension of T to (A # ) b C, then, an easy computation shows that T # is an algebra homomorphism. Moreover, if ja + ffi A j»i A then it follows from Lemma 4.2 that jaj»2i A and hence that jt # (a + ffi A )j»jt (a)j + jffji B» 3I B : Consequently, T # is an order bounded algebra homomorphism. Again, T # is contractive and since T # (I A )=I B, it follows from Proposition 4.1 that T # is positive and, hence, the restriction of T to A b is positive. Now the relatively uniform continuity oft together with the relatively uniform density ofa b in A imply that T is positive on A and, hence, that T is a lattice homomorphism, which is the desired result. Λ Remark 4.1. Example 5.2 of [3] shows that the condition that T is order bounded cannot be dropped in Theorem 4.3. Next, we give a characterization of algebra isomorphisms in terms of contractive operators. This result is, in some respect, an f-algebra version of the well known Nagasawa's result for Banach algebras ([5, Theorem 1]). Theorem 4.4. Let A and B be Archimedean f-algebras with unit elements e A and e B and let T : A C! B C be an order bounded bijection such that T (e A )=e B. The following properties are equivalent: (i) T is an algebra isomorphism; (ii) T is a lattice isomorphism; (iii) T is bicontractive.

8 30 A. Triki Proof: (i),(ii). Follows from Theorem 4.3 and ([3]; Theorem 5.4). (ii))(iii). Obvious. (iii))(ii). Since T is contractive and T (e A ) = e B, it follows from Proposition 4.1 that T is positive. T 0, the restriction of T to A b, is a bijection onto B b, so, T 0 1 is an order bounded and contractive operator on B b such that T 0 1 (e B )=e A. It follows again from Proposition 4.1 that T 0 1 is positive and, hence, T 0 is a lattice homomorphism. Now, let a and b be elements in A such that a ^ b = 0 and define the sequences a n and b n by putting: a n = a ^ ne A and b n = b ^ ne A for n =1; 2;::: : a n and b n are elements of A b and converge relatively uniformly respectively to a and b. Moreover, we have that a n ^ b n = 0, so, T (a n ^ b n )=T 0 (a n ^ b n )=0: The relative uniform continuity of T now implies that T (a ^ b) = 0. This shows that T is a lattice isomorphism and we are done. Λ Assume now that A and B are Archimedean almost f-algebras and that B is semiprime. Let T : A C! B C be an order bounded algebra homomorphism. N(A), the set of all nilpotent elements in A, is a relatively uniformly closed algebra ideal and order ideal in A. It follows that A=N(A) is a semiprime Archimedean f-algebra. Let s : A C! (A=N(A)) C be the canonical surjection. Then s is a lattice and algebra homomorphism. The linear operator T 0 :(A=N(A)) C! B C defined by T 0 (s(a)) = T (a) for all a 2 A C is an order bounded algebra homomorphism from (A=N(A)) C into B C. So, by Theorem 4.3, T 0 is a lattice homomorphism and hence T = T 0 ffi s is a lattice homomorphism since s is a lattice homomorphism. This leads to the above mentioned generalization of Scheffold's result (cf. [8, Theorem 2.2]) which we state as a theorem. Theorem 4.5. Let A and B be Archimedean almost f-algebras. Assume, in addition, that B is semiprime. If T : A C! B C is an order bounded algebra homomorphism then T is a lattice homomorphism. References [1] Beukers F., Huijsmans C.B., Pagter B. de, Unital embedding and complexification of f-algebras, Math. Z. 183 (1983), [2] Huijsmans C.B., Pagter, B. de, Averaging operators and positive contractive projections, J. Math. Anal. Appl. 113 (1986), [3] Huijsmans C.B., Pagter B. de, Subalgebras and Riesz subspaces of an f-algebra, Proc. London Math. Soc. (3) 48 (1984),

9 On algebra homomorphisms in complex almost f-algebras 31 [4] Luxemburg W.A.J., Zaanen A.C., Riesz Spaces I, North Holland, Amsterdam, [5] Nagasawa M., Isomorphisms between commutative Banach algebras with an application to rings of analytic functions, Kodai Math. Semin. Rep. 11 (1959), [6] Quinn J., Intermediate Riesz spaces, Pacific J. Math. 56 (1975), [7] Schaefer H.H., Banach Lattices and Positive Operators, Springer, Berlin, [8] Scheffold E., FF-Banachverband algebren, Math. Z. 177 (1981), [9] Zaanen A.C, Riesz Spaces II, North-Holland, Amsterdam, Département de Mathématiques, Faculté des sciences de Tunis Campus Universitaire, 1060 Tunis, Tunisie (Received February 8, 2001)